The term Jews can refer to all people who, according to the Tanakh, are descended from Jacob or Israel or only to the descendants of Judah, who was one of the sons of Jacob.  Converts to Judaism and their descendants are also considered Jewish.
A Jew is someone who
- is born Jewish (Defined by halachah, or Jewish Law, as someone whose mother is Jewish; Reform Jews see children of Jewish fathers as well as mothers, who accept a Jewish identity, as Jewish), or
- practices the religion of Judaism (This definition is controversial and not accepted by many Jews.),
- has converted to Judaism.
Jew is a noun, referring to the person. Jewish is an adjective. Judaism is the religion.
A born-Jew or Jew by birth is seen, halakhically, as someone with a Jewish mother. Regardless of whether s/he was raised with a Jewish identity and education, s/he is required to keep the mitzvot prescribed to them in the Torah.
In its loosest sense, practised by Reform Jews, a Jew is someone with one Jewish parent who adopts a Jewish identity.
A born-Jew who was raised secular, but becomes religious, is known as a baal teshuvah.
Simply practicing Jewish traditions, celebrating the holidays or saying the prayers is not sufficient to be accepted by most Jews as making you Jewish. Generally, a formal conversion is required for non-Jews to become Jews.
Another term for someone who converted to Judaism (as opposed to being born Jewish) is Jew by choice.
There is an evolving controversy over conversions: that is, which conversions are accepted by each community.
Orthodox Jews believe that halakhah does not change and all of it is binding. Thus, any conversion not adhering to strict halakhic standards (namely, immersion in a mikveh, acceptance of all of the mitzvot, and a circumcision for males) is not valid. Conversions done by any Orthodox Rabbi are generally seen as kosher in the eyes of Conservative and Reform communities.
Conservative Jews believe that halakhah is binding, but that it can change over time. Conservative Judaism permits driving on Shabbat, which is a violation of not kindling fire on Shabbat. However, Conservative Judaism requires that a male convert undergo circumcision and both female/male converts immerse in a mikveh.
Reform Jews believe that some of the traditions, interpretations and rules that have been developed over time are not binding, so they accept conversions that are based on other steps such as learning about Jewish tradition and law. The ritual of a mikveh is not always required.
It is seen as contemptuous of the beit din to not be fully honest or withhold important information(e.g., motives, saying that a mitzvah does not apply, or if one does not intend to remain living in their denomination's Jewish way, etc.). If one plans to convert to a certain denomination, they should be fully willing to accept their community's requirements and expectations because after they emerge from the process, they are as Jewish as a born-Jew.
The Hebrew word for a convert is גיור, or stranger. The process of conversion is known as גיורות.
Once someone successfully converts to Judaism, s/he is seen as Jewish, period. It is against halakhah to shame a convert. Jews may marry converts, although again, some denominations, such as the Orthodox practitioners will not recognise non-Orthodox conversions and will require a kosher conversion. A kohein (translated as king (?) or priest) may not marry a convert at all, regardless of whether s/he has had a halakhic conversion. This is due to a kohein's special status in the Jewish world, especially in the times of the Temple, so they are required to be held to a higher standard. Koheinim often have certain last names, such as Cohen.